Least-squares minimization is ubiquitous and a typical implementation involves minimizing a cost function of the form cost(x) = norm(model(x) - data)^2

However, norm(x)^2 is not auto-differentiable at zero because of the sqrt in the implementation of norm.
See: https://juliadiff.org/ForwardDiff.jl/stable/user/limitations/

It would be convenient for LinearAlgebra to provide the square of the 2-norm
as a built-in exported function to encourage use of the auto-differentiable version, e.g.:
norm_sqr(x) = sum(abs2, x)
Importantly, this version is also more efficient because it avoids the redundant sqrt and squaring operations.

The predictable pushback here is that it is a simple one-line function that anyone can implement themselves. My counter is that Julia base provides abs2(x::Real) = x*x (and something related for complex values), presumably both for efficiency and for differentiability compared to the "obvious" way abs(x)^2.

A main question would be the name. A natural parallel to abs2 would be norm2 but that name is already used internally in the package and could be confused with the 2-norm itself.

The package already has norm_sqr hidden inside it:

but it is not exported, there is no docstring, and sadly it uses the inefficient and non-differentiable version!

Proposal:

• fix the definition to be norm_sqr(x) = sum(abs2, x) or norm_sqr(x) = sum(abs2 ∘ float , x)
• add docstring
• export it.

I'd make a PR if there is openness to this.

I am unsure whether the ∘ float part should be included. I saw that version in #1650